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Thread: Another counting problem

  1. #1
    MHF Contributor alexmahone's Avatar
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    Another counting problem

    How many 2 x 2 matrices are there with entries from the set {0, 1, ..., k} in which there are no zero rows and no zero columns.

    My solution

    No. of matrices in which there are zero rows$\displaystyle =2(k+1)^2-1$ (Assume that the first row is a zero row. There are $\displaystyle (k+1)(k+1)=(k+1)^2$ ways of choosing the elements of the second row. Since the second row could as well be the zero row, we get $\displaystyle 2(k+1)^2$ ways. Since we have counted the zero matrix twice, we must subtract 1.)
    No. of matrices in which there are zero columns$\displaystyle =2(k+1)^2-1$
    No. of matrices in which there are both zero rows and zero columns$\displaystyle =4k+1$ ($\displaystyle 4k$ matrices each with one non-zero element and 1 zero matrix.)
    No. of matrices in which there are either zero rows or zero columns$\displaystyle =2[2(k+1)^2-1]-(4k+1)=4k^2+4k+1$
    Total no. of matrices$\displaystyle =(k+1)^4$
    No. of matrices in which there are no zero rows and no zero columns$\displaystyle =(k+1)^4-(4k^2+4k+1)=k^4+4k^3+2k^2$

    Could someone please check if my solution is correct? Thanks.
    Last edited by alexmahone; Dec 19th 2011 at 06:55 AM.
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  2. #2
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    Re: Another counting problem

    Quote Originally Posted by alexmahone View Post
    How many 2 x 2 matrices are there with entries from the set {0, 1, ..., k} in which there are no zero rows and no zero columns.
    No. of matrices in which there are no zero rows and no zero columns$\displaystyle =(k+1)^4-(4k^2+4k+1)=k^4+4k^3+2k^2$
    That is the correct answer. I will show you another way to think of it.
    The matrix pattern $\displaystyle ~\left(\begin{matrix} x & x\\x & x\end{matrix}\right)$ where $\displaystyle x$ is not zero can be formed in $\displaystyle k^4$ ways.

    The matrix pattern $\displaystyle ~\left(\begin{matrix} 0 & x\\x & x\end{matrix}\right)$ where $\displaystyle x$ is not zero can be formed in $\displaystyle k^3$ ways, but there 4 of those patterns.

    The matrix pattern $\displaystyle ~\left(\begin{matrix} 0 & x\\x & 0\end{matrix}\right)$ where $\displaystyle x$ is not zero can be formed in $\displaystyle k^2$ ways and there are 2 of those patterns.

    That gives the same answer.
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